Optimal. Leaf size=222 \[ -\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {310399 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725} \]
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Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164,
114, 120} \begin {gather*} -\frac {310399 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2108 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{1701}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {64628 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{8505}-\frac {310399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{76545} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4}{405} \int \frac {\left (\frac {1625}{4}-\frac {13175 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\left (\frac {1409075}{8}-\frac {1211775 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42525}\\ &=\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (\frac {2027475}{8}-\frac {23279925 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{637875}\\ &=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\frac {53475825}{16}+\frac {1883325 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5740875}\\ &=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {25111 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{382725}+\frac {3414389 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{765450}\\ &=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {310399 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}\\ \end {align*}
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Mathematica [A]
time = 8.09, size = 112, normalized size = 0.50 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (21964+245751 x-259650 x^2-386100 x^3+567000 x^4\right )}{\sqrt {2+3 x}}+50222 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+10192945 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{2296350} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 153, normalized size = 0.69
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-170100000 x^{6}+10243167 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-50222 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+98820000 x^{5}+140508000 x^{4}-100684800 x^{3}-37330230 x^{2}+21458670 x +1976760\right )}{2296350 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {200 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81}-\frac {5660 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701}+\frac {1850 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701}+\frac {26417 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{76545}+\frac {713011 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3214890 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {25111 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1607445 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \left (-30 x^{2}-3 x +9\right )}{729 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 43, normalized size = 0.19 \begin {gather*} \frac {{\left (567000 \, x^{4} - 386100 \, x^{3} - 259650 \, x^{2} + 245751 \, x + 21964\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{76545 \, \sqrt {3 \, x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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